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Asymptotic Growth of $(-1)^{r} Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ and the Reverse Higher Order Turán Inequalities for $\sqrt[n]{\overline{p}(n)/n^α}$

Gargi Mukherjee

Abstract

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of $\sqrt[n]{\overline{p}(n)/n^α}$ for $α$ being a non-negative real number, namely $(-1)^{r}Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ by presenting an inequality of it with a symmetric upper and lower bound. Consequently, we arrive at log-convexity of $\sqrt[n]{\overline{p}(n)}$ and $\sqrt[n]{\overline{p}(n)/n}$, previously studied by the author. The another main objective of this paper is to introduce the notion of the reverse higher order Turán inequalities and we prove this for $\sqrt[n]{\overline{p}(n)/n^α}$, which not only generalize the study of Sun, Chen, and Zheng but also depicts the non real-rootedness of the Jensen polynomial associated with the sequence mentioned before.

Asymptotic Growth of $(-1)^{r} Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ and the Reverse Higher Order Turán Inequalities for $\sqrt[n]{\overline{p}(n)/n^α}$

Abstract

Let denote the overpartition function. In this paper, we study the asymptotic growth of finite difference of logarithm of for being a non-negative real number, namely by presenting an inequality of it with a symmetric upper and lower bound. Consequently, we arrive at log-convexity of and , previously studied by the author. The another main objective of this paper is to introduce the notion of the reverse higher order Turán inequalities and we prove this for , which not only generalize the study of Sun, Chen, and Zheng but also depicts the non real-rootedness of the Jensen polynomial associated with the sequence mentioned before.
Paper Structure (8 sections, 17 theorems, 166 equations)

This paper contains 8 sections, 17 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Inequalities for $(-1)^r \Delta^r \log r_{\alpha}(n)$
  3. Preliminary lemmas
  4. Proof of Theorem \ref{['mainresult2']}
  5. Reverse higher order Turán inequalities for $r_{\alpha}(n)$
  6. Upper bound of $u_{\alpha}(n)$
  7. Proof of Theorem \ref{['mainresult3']}
  8. Conclusion

Key Result

Theorem 1.5

Let $N(r,\alpha)$ be as in def8. For $r \in \mathbb{Z}_{\geq 2}$ and $n \geq N(r, \alpha)$, where $C(r)$ and $C(r,\alpha)$ are defined in def3 and def4 respectively.

Theorems & Definitions (32)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Remark 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 22 more